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tut5large - ∪ A ∩ C 6 Using the laws of set theory...

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Problems to Week 6 Tutorial — MACM 101 1. Prove or disprove each of the following: (a) For sets A,B,C U , if A C = B C , then A = B . (b) For sets A,B,C U , if A C = B C and A C = B C , then A = B . 2. Prove that A - B = A B . 3. Prove that A Δ B = ( A B ) ( A B ). 4. Investigate the truth or falsity of the following using 3 methods: Venn diagrams, laws of set theory, and proving that the left side is a subset of the right one and vice versa. A - ( B C ) = ( A - B ) ( A - C ) . 5. Use the laws of set theory to establish that ( A B ) ( A C ) = ( A B
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Unformatted text preview: ) ∪ ( A ∩ C ) . 6. Using the laws of set theory, simplify each of the fol-lowing (a) ( A ∩ B ) ∪ ( A ∩ B ∩ C ∩ D ) ∪ ( A ∩ B ), (b) ( A-B ) ∪ ( A ∩ B ). 7. Construct the power set of the set {∅ , { 1 } , {{ a }}} . 8. Let A,B,C,D be nonempty sets. Prove that A × B ⊆ C × D if and only if A ⊆ C and B ⊆ D . 9. Prove that A × ( B-C ) ⊆ ( A × B )-( A × C ). Does the equality holds? 1...
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